godel

Gödel's incompleteness theorem

In any formal system S, which is consistent, there can be a proposition which denies the provability of that proposition (of itself) within the system; i.e., the statement this statement cannot be proven within S can exist within S. Since this proposition can exist then it must be true, which denies its 'not provable' status, and therefore produces an inconsistency within what is supposed to be a consistent system. Thus no formal system of propositions can be complete. 

Gödel 1931

[ Or, more simply:

There exist numbers and functions that cannot be computed by a logical (Turing) machine. ]

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Current scientific domains politely ignore the rather significant implications of there being limits to logic itself. Gödel's incompleteness theorems (above) suggests that any formal system (i.e., a rule-based system of thought such as logic, mathematics, language, physical theory and ecological theory) contains self-referential paradoxes (statements whose truth is undecidable within the rules of that system):

That is, it is not possible to prove that the rules (the accepted body of laws and theories) of a formal system of knowledge is self-consistent using the very same rules: the circular, self-referential loops bounds the formal system!

The importance of this theorem lies in the fact that an inconsistent system of knowledge is usually considered to be poor science: one can prove anything one likes in an inconsistent system. That scientific theories are logically inconsistent is a particularly troublesome blow to the "sanctity" of the Scientific Method.

For example, the principle of Competitive exclusion is a central tenet of ecological thought. However, even when it was experimentally shown that competitive exclusion does not always happen (e.g., for Drosophila sp. by Ayala 1970), the competitive exclusion principle was defended with the argument that the organisms tested had different niches and so did not represent a valid test (Gause 1970). This latter statement demonstrates the intrinsic circularity (self-referential nature) of the ecological concept of competitive exclusion. If competition is not observed, it is because the organisms have different niches (Hardin 1960, McIntosh 1985:186). Thus, it is impossible to prove or disprove the importance of competition, using the formalisms associated with principle of competitive exclusion - some other external (independent and more general) principle must be called upon to prove or disprove it.

Another such example is Optimal foraging theory: a given currency is being optimised by an organism. If the currency is found not to be optimised, that is because the choice of the currency is not appropriate or correct. The correct currency is the one that is optimised! Another famous example of (paradoxical) circularity of thought revolves around the most fundamental of biological principles, the concept of Natural Selection. The formalism is that what is fittest will survive and reproduce. What is fittest is that which is most adaptive. And that which is most adaptive is that which is most fit to survive. Again, it is impossible to prove what is fit and what is not because fitness and survival refer to (and define) each other.

Another example of what may be considered such a circular concept is the very notion of ecological succession/climax: the observation that vegetational systems coherently change from some locally disturbed state (fire, storms) to some more regional stable state through a series of recognizable stages (i.e., larger spatial scales driven by climatic and evolutionary processes). Note the same difficulty: What is a climax state? A state to which a disturbed state evolves into. What is a disturbed state? A state which will change into a climax. Perhaps for (I hope obvious) obvious reasons, it has not been possible to confirm or refute the presence of ecological succession since the inception of the concept in the early 1900s (Tansley, Gleason, Wittaker), using the formal language and concepts internal to ecology.

Reacting to some of the confusion and verbosity caused by this intrinsic circularity of ecological ideas, some have gone so far as to suggest that they serve nothing but to confuse the real issues at hand, which should be the development of a predictive, empirical science (e.g., Rigler 1975). However, intrinsic circularity does not mean that we must abandon all attempts at communication and comprehension; rather it means that axiomatic systems cannot be proved or disproved to be consistent within the formal logic of that system. That is, ecological concepts cannot be proved or disproved with ecological arguments. However, even though they may stand upon shaky theoretical foundations, many of these axiomatic systems function to some extent (though many may disagree), conveying information and serving a useful, albeit, imperfect function.

To prove the existence of optimal foraging or natural selection, one must go beyond the confines of each formalism. To prove the existence of ecological succession, one must go beyond the confines of ecological formalisms. But how? The question being, what is this meta-formalism? If proof or justification of a theory (e.g., some Grand Unified Theory) is required one must look to some more superseding set of principles (which by definition nullifies the grand-unifiedness of the original theory), with the implication that these superseding principles are themselves also bound to the very same limitations of logical incompleteness. For example, consider the progression from the world visions of Aristotle to Copernicus to Galileo to Kepler to Newton (Classical Mechanics) to Einstein (theories of Special Relativity to General Relativity) to Witten (String theory to M-theory), where each body of knowledge and rules attempts to go beyond the constraints of the former. However, when a degree of consistency across so many superseding (hierarchical) sets of formal systems are observed as in this example, then one may have some reason to suggest that concepts and principles are convergent upon some meta-stable set of quasi-self-consistent formalisms. 

Not a satisfactory answer I am sure. But hopeful, nonetheless. This is the hope and also the cause of my interest in Thermodynamic Theory. Most people at least do agree to the direction of time.